Theorem: Finite promised-table realization

Fix integers $p_{\mathrm{rep}}\ge 0$, $p_{\mathrm{clk}}\ge 1$, $q\ge 6$, $w_{\mathrm{src}}\ge 1$, and $n\ge 1$ original matter names, including the named source qubit.

On the canonical promised forward domain of Definition def:forward-domain, with valid incoming locations and clean name history, the table step has an exact NOT–CNOT–Toffoli realization $U$ using

classical bits, excluding internal matter and external references, and at most

gates.

Clean synthesis work returns to zero. Internal matter, its external references, and valid location words are unchanged. Complete geometric input history is retained. Writing $U$ also for its identity extension to matter, the decoder satisfies

on this domain, with $\kappa =\theta =1$ and the fixed table metadata.

Proof. Apply Lemma lem:synthesis to $P_{\mathrm{geo}}$ with $N=2B$. It realizes (eq:forward-map) on the forward domain and restores $2B-2$ work bits.

The promise gives

The decoder therefore reconstructs exactly the geometry and certificate of Theorems thm:persistence and thm:source-map. Exact algebraic floors provide the canonical output reports. The parameter stays at $s=-1/2$, so valid incoming abstract cell labels remain valid after packet replacement.

For each name, copy each of its six location bits into its clean name-history bit, controlled by the output stage bit. Each copy is one Toffoli with stage and location as controls, using $6n$ gates. Locations are unchanged and no name bit influences a geometric output.

There are $2B$ logical geometric/history bits, $2B-2$ work bits, and $12n$ location/history bits, giving (eq:total-bits). The synthesis bound and name copies give (eq:gate-bound). No gate acts on internal matter or references. The decoder ignores implementation history, proving (eq:decoder).